<h2>题目编号 : 241</h2>
<div style="color:#666;font-size:80%;">18 April 2009</div><br />
<div class="problem_content">
<p>For a positive integer <var>n</var>, let &sigma;(<var>n</var>) be the sum of all divisors of <var>n</var>, so e.g. &sigma;(6) = 1 + 2 + 3 + 6 = 12.
</p>

<p>A perfect number, as you probably know, is a number with &sigma;(<var>n</var>) = 2<var>n</var>.</p>

<p><div style='text-align:left;'>
<table>
<tr>
<td>Let us define the <b>perfection quotient</b> of a positive integer as</td><td><var>p</var>(<var>n</var>)</td><td>=&nbsp;</td>
<td><div style='text-align:center;'>&sigma;(<var>n</var>)<br /><img src='images/blackdot.gif' width='30' height='1' alt='' /><br /><var>n</var></div></td>
<td>.</td>
</tr>
</table>
</div>
</p>
<!--
<p>Let us define the <b>perfection quotient</b> of a positive integer as <var>p</var>(<var>n</var>) = <font "size=4"> <img src="" style="display:none;" alt="^(" /><sup>&sigma;(<var>n</var>)</sup><img src="" style="display:none;" alt=")" />&frasl;<img src="" style="display:none;" alt="_(" /><sub><var>n</var></sub><img src="" style="display:none;" alt=")" /></font>.</p>
-->
<p>Find the sum of all positive integers <var>n</var> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> 10<img src="" style="display:none;" alt="^(" /><sup>18</sup><img src="" style="display:none;" alt=")" /> for which <var>p</var>(<var>n</var>) has the form <var>k</var> + <img src="" style="display:none;" alt="^(" /><sup>1</sup><img src="" style="display:none;" alt=")" />&frasl;<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" />, where <var>k</var> is an integer.</p>

</div><br />
